Single Variable Calc

Some formulas (rules, etc.)

Differential

Study of instantaneous rate of change of functions.
Slope.

What is a function?
A: black box w/ mapping of 1 in to 1 out (not reverse, Vertical Line Test)

$$x\in D,f(x)\in S,f:D\to S$$ $$D\subset \Re ,S\subset \Re ⟹f:\Re \to \Re$$

Note to self: functions work well for events in time, or causally-linked events.
Note note to self: what are causally linked events?

Polynomial

A function of the form $a_n{x}^n+...+a_0,a_i\in R,a_n\ne 0,degree=n$

Intermediate Value Theorem IVT

On any interval of cnts $f(x)$, $f(c)$, where c is in the interval, will be between the values of f at the bounds of the interval.

Min-Max Theorem

Let $f$ be continuous on a closed interval. Then it has an absolute max & min on that interval.

Differentiable at $x_0$

$f$ is diffable at $x_0$ if the following limit exists and is finite:

$$\underset{x\to x_0}{\lim }\frac{f(x)-f(x_0)}{x-x_0}=f^{\prime }(x_0)$$

This means the slope of the chord from the two points converges to a finite number.
Alternatively, there is a ‘good’ approx to $f(x)$ in a nbhd of $x_0$ .

Equivalent Notation

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

Use this definition of a derivative to prove formulas.

Leibniz Notation

$$f^{\prime }(x)=\frac{dy}{dx}=\frac{d}{dx}f(x)=D_{x}f(x)$$

For derivative at $x_0$ write

$$\frac{dy}{dx}{|}_{x=x_0}$$

Some rules

These are generally proven using the h notation plus limit theorems.
$f(x)=c,f^{\prime }(x)=0$
$f(x)=x,f^{\prime }(x)=1$
$(f+g{)}^{\prime }(x)=f^{\prime }(x)+g^{\prime }(x)$
$(fg{)}^{\prime }(x)=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$
$(f\circ g)(x)=f^{\prime }(g(x))g^{\prime }(x)$

All Da Proofs

MVT

If $f$ cts and diffable on $[a,b]$ then $\exists c,s.t.$

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a},a<c<b$$

A policeman stops you and asks to see your toll ticket, which shows that you entered the road 2 hours earlier. The policeman notes that you are exactly 242 km from where you entered. He then gives you a traffic ticket. You protest because you know that in the last several minutes you were driving well under the speed limit. The policeman replies: “Your average speed was 121 km/hour. Therefore by the Mean Value Theorem, at some point in your trip you were traveling at 121 km/hour.”

Generalized MVT

$$\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime }(c)}{g^{\prime }(c)}$$

Derivatives of sums are easier than derivatives of products and the log of a product is the sum of the logs

Local minima/maxima: slope is zero

• Important for optimization!

Linear Approximation

Linearization near a point a:

$$f(x)\approx L(x)=f(a)+f^{\prime }(a)(x-a)$$

In the sense that:

$$\underset{x\to a}{\lim }\frac{f(x)-L(x)}{x-a}=0$$

So $L(x)$ is a tangent line to f at some point a. (Not chord).
Error in approximation:

$$E(x):=f(x)-L(x)=\frac{f^{\prime \prime }(s)}{2}(x-a{)}^2,s\in (a,x)$$

Equivalently,

$$f(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(s)}{2}(x-a{)}^2$$

Where $a$ is some point on domain of $f$ and $s$ is some point between $a$ and $x$.

Weird result, since we are re-defining $f$ in terms of the tangent at some point on it, plus this weird double derivative piece.

The proof uses GMVT followed by MVT.

Corollary:

$$|E(x)|<(K/2)(x-a{)}^2,|f^{\prime \prime }(t)|<K$$

Differentials notation

$\Delta x:=x-a$
$\Delta y=f(a+\Delta x)-f(a)$
Some change in x and corresponding change in f(x).

$dy:=f^{\prime }(a)\Delta x$
Then, $\Delta y\approx dy$

Leibniz notation

$$dy:=(\frac{dy}{dx}{|}_a)\Delta x$$

Taylor Polynomials

$$P_n(x)=\sum _{j=0}^n\frac{f^j(a)}{j!}(x-a{)}^j$$

This is the n-th Taylor polynomial, which is a generalization of linearization to higher degree polynomials.
Assumes $f$ is at least $n$ times diffable at $a$.

Corollary

$$P^{(k)}(a)=f^{(k)}(a)$$

And it is a unique polynomial that satisfies this condition.
Why unique? (Some proof involving re-writing any polynomial in terms of x-a)

We can also iteratively compute $P_n(a)$ , by just adding the new term.

Lagrange remainder

If $f(n+1)(t)$ exists for all $t$ in an interval containing $a$ and $x$, then there exists some $s$ in between $a$ and $x$ s.t.

$$E_n(x)=f(x)-P_n(x)=\frac{f^{(n+1)}(s)}{(n+1)!}(x-a{)}^{n+1}$$

Note similarity of this error to the lin approximation error. Proof is by induction, but takes a bit of effort and algebra…

Big O

Defn: $f(x)=O(u(x))$ as $x\to a$ if there exists $K>0$ and an open interval $I$ containing $a$ s.t. for all $x\in I$

$$|f(x)|<K|u(x)|$$

Simple properties:

1. If $f(x)=O(u(x))$ as $x\to a$, then $Cf(x)=O((u(x))$ for any $C$.

2. If $f(x)=O(u(x))$ and $g(x)=O(u(x))$ as $x\to a$, then $f(x)\pm g(x)=O(u(x))$ as $x\to a$ (use triangle inequality).

3. 3. If $f(x)=O((x-a{)}^{k}u(x))$ as $x\to a$, then $\frac{f(x)}{(x-a{)}^k}=O(u(x))$

$$E_n(x)=O((x-a{)}^{n+1})$$

Equivalently:

$$f(x)=P_n(x)+O((x-a{)}^{n+1})$$

$P_n(x)$ is the unique polynomial of degree at most $n$ that approximates $f(x)$ with error on the order of $|x-a{|}^{n+1}$

Maclaurin Polynomials

:= Taylor polynomials at $a=0$

$$M_n(x)=\sum _{j=0}^n\frac{f^j(0)}{j!}(x{)}^j$$

Q: why are they useful?

Integrals

Defn: An anti-derivative of a function $f$ is a function $F$ s.t. $F^{\prime }(x)=f(x)$

Defn: An indefinite integral of a function $f$ is written:

$$\int f(x)dx=F(x)+C$$

Chapters 5,6,7,8,9 of Single Variable Calc

1. Area under a curve
2. $dA=f(x)\ast dx$
3. Note: $\frac{dA}{dx}=f(x)$
4. Choose smaller and smaller $dx$ to sum the true area under the curve

Reimann sum definition of definite integral

$${\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f(x_i^{\ast })\Delta x_i$$

where the interval $[a,b]$ is divided into $n$ subintervals of equal width $\Delta x_i=\frac{b-a}{n}$ , and $x_i^{\ast }$ is a sample point in each subinterval.

Fundamental Theorem of Calculus

1. First Part: It states that if $F(x)$ is an antiderivative of a continuous function $f(x)$ on the interval $[a,b]$, then the definite integral of $f(x)$ from $a$ to $b$ is given by:

$${\int }_a^{b}f(x)dx=F(b)-F(a)$$

This means that the integral (area under the curve) can be found using the antiderivative.

2. Second Part: It states that if $f(x)$ is continuous on $[a,b]$ and $F(x)$ is defined by the integral $F(x)={\int }_a^{x}f(t)dt$ , then the derivative of $F(x)$ is $f(x)$ :

$$F\prime (x)=f(x)$$

This means that the derivative of the integral function recovers the original function.

Proof of part 1 (visually makes sense), uses reimann sum definition and MVT for integrals:

Integral MVT

For a continuous function $f(x)$ on $[a,b]$, there exists a point $c\in [a,b]$ such that:

$$f(c)=\frac{1}{b-a}{\int }_a^{b}f(x)dx$$